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DIAGNOSTIC  AND  REMEDIAL  TREATMENT  FOR  ERRORS  IN  ARITHMETICAL'  TSA'GONINC 


By 
.7.  J.  Osourn 

Director  oi  induce  tionai  iAeasureuents 


Issued  by 
State  Department  of  Public  Instruction 
Kadi son,  Wisconsin 


December  1922 


01 


DIAGNOSTIC  AND  REMEDIAL  TREATMENT  FOR  ERRORS  IN  ARITHMETICAL  REASONING 

INTRODUCTION 

A  voluminous  literature  nas  appeared  in  recent  years  toased  upon  trie 
correct  reponses  which  children  make.   The  last  decade  will  be  remen^e- 
ed  in  educational  history  as  the  time  when  large  numbers  of  standard 
tests  first  began  to  be  published,  and  when,  the  rank  and  file  of  the 
teaching  profession  began  to  get  aqquainted  with  the  application  of 
elementary  statistical  procedure  to  the  field  of  education*  Splendid  as 
this  movement  has  been  it  is  by  no  means  complete,   The  purpose  of  this 
and  other  circulars  which  are  To  follow  is  to  describe,  in  some  measure. 
a  further  line  of  attack  upon  our  educational  problems  based  oh  the 
study  of  wrong  responses. 

There  is  good  argument  to  support  the  thesis  that  a  study  based  upon 
wrong  responses  is  likely  to  be  even  more  stimulating  and  helpful  than 
a  procedure  based  upon  the  results  of  the  child's  susseesful  efforts. 
In  the  business  wo  aid  much  study  is  given  to  the  points  at  which  the  out 
put  of  the  factory  Jraila  to  function  as  it  should.   The  manufacturers 
of  automobiles,  for  example,  have  '■very  definite  information  concerning 
the  weaknesses  of  their  cars.   From  a  study  of  these  weaknesses  it  is 
possible  to  improve  the  output..  tn   general  a  contact  with  failures  and 
breakdowns  stimulates  thinking-   Improvements  can  come  only  when  there 
is  a  somewhat  definite  knowledge  of  failures. 

Progress  in  the  study  of  errors  has  long  been  retarded  by  the  idea 
that  there  is  the  utmost  aiversity  in  the  mistake  of  children.   The 
actual  evidence  is  far  to  the  contrary,   The  same  wrong  answers  occur 
in  all  types  of  schools  and  over  wide  areas  to  such  an  Sxtent  that  it 
js  possible  to  forecast  how  many  pupils  in  any  individual  school  will 
make  certain  mistakes.  Errors  are  like  diseases  in  some  respects. 
There  are  a  great  many  different  diseases,  but  only  a  few  of  them  are 
wide-spread  and  frequent  in  occurrence. 

In   like  manner  the  evidence  so  far  available  shows  that  four  or  five 
different  things  are  resppnsible  for  at  least  three  fourths  of  the 
trouble  which  children  have.   To  identify  these  widespread  errors, 
determine  their  causes,  and  plan  their  cure  is  clearly  an  important 
task  in  educational  research.  Such  data  will  lead  to  more  intelligent 
supervision,  more  scientific  methods  of  teaching,  better  organization 
of  curriculum  material,  and  more  efficient  teachert~training. 

With  these  facts  in  mind  it  is  rather  surprising  to  find  how  little 
is  available  in  print  concerning  the  nature  and  cause  of  fa: lure.   This 
circular  is  one  of  a  series  begun  last  year  and  designed  to  investigate 
these  questions  as  they  apply  to  the  sever:!  school  subjects, 

THE  IMPROVEMENT  0?  ABILITY  IN  ARITHMETICAL  REASONING 

In  discussing  a  remedial  program  on  Arithmetical  Reasoning  the 
following  items  must  be  known  and  agreed  upon:- 


575378 


-2- 

1.  A  list  of  typical  problems  which  have  been  widely  used  and  on  which 
lists  of  wrong  answers  arc  available. 

2.  An  identification  and  classification  of  the  most  serious  disabili- 
ties. 

3.  A  list  of  the  wrong  answers  which  are  symptomatic  of  each  disability 
in  each  type  of  problem^.. 

4.  Tre  remedy  to  apply  after  the  specific  disabilities  are  known  and 
indentified. 

Each  of  the  these  will  be  considered  in  order. 
1.  A  list  of  the  Problems  which  were  used. 

In  this  study  the  problems  in  the  Buckingham  Problem  Test,  Porm  1 
were  used. 

{l)  V7e  learn  2  words  a  day  in  our  class.  How  many  do  we  learn  in  8 
days? 

($}}23  children  belong  to  our  class  but  only  19  are  present.   How  many 
children  are  absent? 

(3)  James  has  28  marbles.   He  gives  half  of  them  to  Charles.   How  many 
has  he  left9 

(4)  If  you  can  get  3  ginger-bread  dogs  for  5  cents,  how  many  can  you 
get  for  ten  cents? 

(5)  A  boy  owned  3  kites,  each  of  them  having  150  feet  of  string.  How 
many  feet  of  string  had  he? 

(5)  A  baseball  team  took  12  players  on  a  trip.  The  trip  cost  the  team 
$36.00.   How  many  was  that  for  each  player? 

(7)  An  automobile  was  run  30  miles  every  day  for  a  week.   How  many 
miles  did  it  go? 

(8)  Henry  gathered  5  quarts  of  nuts.   He  sold  them  at  8  cents  a  quart 
and  spent  the  money  for  oranges  at  4  cents  apiece.   How  many  did 
he  buy? 

(9)  If  an  electric  car  runs  9  miles  an  hour,  how  many  hours  will  it 
take  to  travel  from  one  city  to  another,  117  miles  away? 

( 10) Ned  sold  his  rabbit  for  30  cents.   This  was  3/5  of  what  he  paid. 
What  did  he  pay  for  the  rabbit? 

(ll)lf  a  girl  had  two  one-dollar  bills,  three  five-cent  pieces,  two 
dimes,  and  three  quarters,  how  much  money  did  she  have? 

( 12) now  many  pencils  can  you  buy  for  50  cents  at  the  rate  of  2  for  5 

cents? 

(13)A  boy  had  210  marbles.  He  lost  l/3  of  them.  How  m^.ny  were  left? 

(14) Two  tubs  of  maple  sugar  weighed  42  lbs.  One  weighed  18  l/4  lbs. 

How  many  pounds  did  the  other  weigh? 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/diagnosticremediOOosburich 


-3- 

(16)  A  store  takes  in  the  following  suns:  $1250;  v-00,  $175,  $16^.25.,^ 
^120.50,  $32.75,  $68.50.   It  pays  out  $600,  <i360,  £165.67,  #33.3v3 
$240.   How  much  remains  after  payments  are  made? 

(I6j  A  man  bought  a  house  for  $7250,  After  spending  $321.50  for  re- 
pairs, he  sold  it  for  $9125.   How  much  did  he  ga:.n? 

(17)  How  many  weeks  will  it  take  Joseph  to  same  21  dollars  for  a 
bicycle  if  he  saves  1  1/2  dollars  a  week? 

(18)  George  walks  at  the  rate  of  2  l/3  miles  per  hour.  Hcnrj  starts 
with  him  and  walks  in  the  same  direction  at  the  rate  of  3  miles 
an  hour.   How  many  miles  apart  will  they  ha  in  3  hours? 

(19)  If  0.  78  of  the  weight  of  potatoes  is  water,  how  many  pounds  of 
water  are  there  in  a  bushel  of  potatoes  if  a  bushel  of  potatoes 
weighs  60  pounds? 

(20)  Two  boys  agreed  to  cut  a  lawn  for  9C  cents.  The  first  boy  worked 
one  hour  and  the  second  boy  worked  4  hotirs.  How  much  money  should 
each  receive? 

(21)  Find  the  total  cost  of  the  following  purchases:  8  l/2  yds  flannel 
at  96  cents,  4  3/4  yds  braid  at  16  cents..  12  yds.  enbroidery  at 
22  1/2  cents,  10  yds.  ic.ce  at  27  1/2  cants. 

(22)  The  distance  from  Hew  York  to  Chicago  is  993  miles.   The  running 
time,  by  the  Hew  York  Central  is  1°  hours.  j3ocause  cf  floods, 

an  engine  goes  only  295  miles  the  first  7  hours.   After  that  how 
many  miles  per  hour  must  it  go  to  reach  Chicago  Oxi  time? 

(23)  A,  S,  and  C  owned  all  the  stock  in  a  certain  store  worth  $4,300. 
A  owned  16  shares,  B  5  shares,  and  C  5  shares.  What  was  the 
money  value  of  each  person's  share  of  the  stock? 

(24)  A  real  estate  dealer  made  a  profit  of  $5,000  on  a  plot  cf  ground. 
The  rate  of  profit  was  20/.  j'md  the  cost  of  the  plot. 

(25)  It  is  estimated  that  if  the  nouse  fly  were  Kept  from  food,  deaths 
in  cities  in  summer  would  be  1/3  less.  If  the" deaths  in  the 
summer  months  are  at  the  rate  of  12  per  1  ,.000  people,  how  many 
lives  might  thus  be  saved  in  one  summer  in  a  city  of  1,000,000 
inhabitants? 

(26)  I  bought  a  cask  of  molass-es  containing  84  gallons  for  $28.  Hine 
gallons  having  leaked  out,  at  what  price  per  gallon  must  I  sell 
the  remainder  to  ^  a  in  ^4^.25? 

(27)  A  contractor  completed  2/5"  of  a  job  in  12  l/2  days.   How  many 
days  longer  should  it  .take  to  finish  the  job? 

(28)  In  the  manufacturing  of  jute  twine,  the  raw  jute  costs  8  3/5 
cents  a  pound,  the  spinning  cost  7/8  of  a  cent  a  pound,  and  the 
twisting  and  finishing  cost  15/16  of  a  cent  a  pound.   Find  the 
cost  of  1,500  pounds  of  jute  twine. 

(29)  On  a  map,  a  line  2  3/4  incnes  long  represents  a  distance  of  132 
miles.  How  many  miles  are  represented  by  a  line  8  l/2  inches  long' 


-4- 

(30)  Mr.  Wood  had  250  baseballs.   He  sold  20$  of  them  tne  first  day 
ana  40^  oi  tne  remainder  the  second  day.  what  per  cent  o,  his 
original  stock  was  left? 

2.   Identification  and  Classification  of  the  moat  serious  disabilities 

The  following  classification  and  nomenclature  is  tentative  only. 
It  is  based  on  a  study  of  about  30,000  errors  made  by  6000  children  in 
eighteen  counties  and  one  large  city.   The  percents  shew  how  frequently 
each  type  of  error  occurred.   The  classification  is  as  follows :- 

Approximate 
Frequency 

(1)  Total  failure  to  comprehend  the  problem... 30$ 

(2)  Procedure  partly  correct  but  with  the  omission  of  one 

or  two  essential  elements 20% 

(This  type  will  be  designated  as  partial  response 
hereafter) 

(3)  Ignorance  of  fundamental  quantitative  relations  10^ 

Total  60$ 

(4)  Errors  in  fundamentals  20$ 

(5)  Miscellaneous  errors  2$ 

(6)  Errors  whose  cause  coald  not  be  discovered  .  ....  18$ 

Total  100$ 

$.    Symptoms  of  the  Several  Disabilities. 

When  tests  have  been  given  and  the  results  are  poor,  the  super- 
visors and  teachers  will  want  to  know  how  to  tell  which  disability  is 
causing  the  trouble. 

In  the  following  summary  a  list  of  the  more  serious  disabilities 
is  given  together  with  z he  wrong  answers  which  are  the  symptoms  of 
each,   when  a  pupil  gets  a  wrong  answer  to  a  problem  in  this  test  look 
for  the  wrong  answer  in  the  list  which  follows.   If  it  appears  here 
the  name  of  the  disability  which  it  Represents  will  rlso  be  given. 

The  Typical  Wrong-  Answers  And  T he  .  Disabilities  of  Which  They 

Are  '£h£S7/m-,3toms. 

Problem    Y/rong  Answers  Disability 

(l)       10,  4,  6,  Total  failure  to  comprehend 

18,14,  15  Errors  in  the  fundamentals 

1 8,  2 Partial  response    


-5- 


Problem   Wrong  Answers  Pisafrilfcty 

(&)      42,  437,  1  4/19  Total  failure  to  comprehend 

"5  14,  16,  5,  6 ?:IT£X§- J-IL. -£^.nJl'^Bg-n1'' al  s 


Err  or  f.  in  f  ■3r..:3ar.entals 


(3)  13.  18,  24,  12.  15,  16 

(4)  18,  15,  50             Total  failure  to  comprehend 
2,  3Q, Partial  : 'Q g  ponse ._ 


(5)         153 f  50                Total  failure  to  comprehend 
453,  350,  550.  553,  250 Errors Jn_  fundamentals 


(6)     1^432,  $24,  f-48  Total  failure  to  comprehend 

$2,  $4,  $5,  $6   Error s  i n  fixs^cxmn  ta3  s 


(7)     180,  150  120,  160  Ignorance  of  quantitative  relations 

100  Total  failure  to  comprehend 

217  Errors  in  fundamentals 

30,  7 Partial  response ; 


(3)         17,  32,  20  Total  failure  to  comprehend 

2,  40.  5..  8,  4             Partial  response 
11_4 Errors  in  fundamentals 

(9)  1053,  126,  108            Total  failure  to  comprehend 
10,  11,  14,  19 Errors  In  fundamentals 

(10)  38,  30  3/5,  15,  18,  90.  6  Total  failure  to  comprehend 
20,  61,50,  30,  10 Partial  _ r c ££ o  ng e 


(11)     $2.00,  $1.10,  $2.85        Partial  response 

$3.00,  $   3.20,  $3.05,  $2.10 
#1.12  Errcr3  in  fundamentals 


(12)  40,  125,  10,  25,  250         Total  failure  to  comprehend 
100,  30 Partial  response 

(13)  209,2/3,  630,  105,  207         Total  failure  to  comprehend 

70              •       Partial  response 
130,  137 Errors  in  fundam en tals  


-6- 

ProMeni  '   Wrong  Answers.  Disability 

(14)     766  1/2,  60  1/4,  2  22/73,  21   Total  failure  to  comprehend 

24  1/4,  24  3/4,  34  3/4,  34,  23,  1/2 ,) 

)  Errors  in  fundamentals 

34  1/4,  24  1/2  ) 

42,  i  e  1/4 Partial  response ; 


(15)  $1563,  $57?,  $562,80,  $1463,  $463,) 

i  Errors  in  fundamencals 

$562,  S&63.25,  $573,  $553,  $43.25  I 
$1963 ' Partj  al  response 

(16)  $1875  .    Partial  response 
$16696.50  Total  failure  to  comprehend 

$1453.50,  $JA93';50,  $1554  .  50  ,$463,  ) 

$553.' 50,  $302.75  )   Errors  in  f  undauentals 

. 2196.50 Ignorance  of  quantitative  relations 

(17)  31  1/2,  22  1/2,  19  1/2,  20,  63      Total  failure  to  comprehend 
2,  10  1/2,  10,  21,  7,  42.  Partial  response 

13,  15 ; Errors  in  fundamentals 

(18)  1,  6,  8  1/3,  1  1/3,  3  1/3,  2  4/7,) 

21 ,  2/3 ,  5  1/3             ) 
7  ,  9,2  1/3  ,  5 _ Partial  response 


Total  failure  to  comprehend 


(19)  .18,  1.3,  7$,  4680;  138           Total  failure  to  comprehend 
46.7,  46.36,  47.58,  44,8,  43.8 Errors  in  fundamentals . 

(20)  $4.50;    .90    and   $3.60;    .18;    ,20;       ) 

.22  l/2   and    .67   l/2;    .90;    $3.60      )        Total   failure    to   comprehend 

.45 Partial    response 


(21)  $14.21,  $14.23,  $14.52,  $14.39,  ) 

$16.11  )Errors  in  fundamentals 

m ___ ^13.61,  $1.62 Partial  response 

(22)  702,  52  12/19  •  Partial  response 

42.  2/7,  3  55/148  Total  failure  to  comprehend 
63  9/11 Errors  in  fundamen tal s 


-«-*_=-  t 


-7~ 

Problem    Wrong  Answer  -uioaoil  ity_ 

f.23)    $300,  $$600  and  $960  Total  failure  to  comprehend 

$200;  #768,  $144  and  $240        Partial 'response 

$3200,  $600  and  #9090 Errors  in  fundamentals 

(24   $1000,  $6000,  $10000,  $100000, 

$5020,  $4000  Total  failure  to  comprehend 

$550000,  $25000*  .  $250 Err  or  3  in  fundamentals 

(25)   12000,  8333  l/3,  1000,  3,996000, 

1200  Partial  response 

333  1/3,  666  2/3,  3333  l/3,  66666  2/3  Failure  to  comprehend 

40000,  400,  4000000 Err on r  p  \ i a  f tadaiae nt ala 


(26)    39,  $32,25  .06 Jte r fc ia  1  response 


(27)   5,  31  1/4,  50,  7  l/2            Total  failure  to  comprehend 
12  1/2,  37  l/2,  25 Zi^hASi-JCSSIiSE^ 


(28)   $150,  $  150C0,  $156.18  3/4,        ) 

$155.93,  $156,11  1/4,  $156.3.0  1/4,3  Errors  in  fundamentals 
3155.  96  ,  $156 } 


(29)      353,  396                      Total  failure  to  comprehend 
1122,  132,  528 Partial  response 


(30)    20  ,  190,  100  Total  failure  to  comprehend 

I 40t__50t_6O,  123  Partial  respcnsS- 

General  Symptoms 

The  disability  which  I  have  called  total  failure  to  comprehend, 
is  indicated  in  general  by  the  fact  that  tne  child  merely  juries  with 
the  figures.  If,  in  any  problem,  his  answer  is  gotten  by  some  aimless 
combination  of  some  or  all  the  numbers  in  the  problem,  it  is  safe  to 
assume  that  he  is  suffering  from  total  inability  to  comprehend. 

Partial  response  means  that  the  child  overlooks  or  fails  to 
respond  to  parts  of  the  problem.  For  example,  in  the  Problem  -"A 
boy  had  219  marbles  and  lost  1/3  of  than.   How  many  had  "he  left?" 
An  answer  of  70  shows  that  the  pupil  overlooked-  or  failed  to  respond 
properly  to  the  words  had  he  left. 

'■-here  the  answer  is  wrong  in  only  a  single  digit,  or  where  the 
decimal  point  is  placed  wrongly  or  emitted,  it  is  sage  to  assume  that 
the  child  has  failed  because  of  a  mistake  in  the  fundamentals.   In  a 
large  number  of  cases  the  latter  type  of  errci  ecmbines  with  one  of 
the  two  proceeding  types  so  that  the  child  gets  an  answer  only  simi3a 
to  those  listed  under . the  first  tw»  types.   In  this  report  these 
errors  are  reported  under  types  1  or  2  only  and  not  under  type  3 


-8- 


4.  Remedial  Treatment, 


Some  of  the  principles  of  Successful  Remedial  Treatment  have 
already  been  suggested.   It  is  essential  that  we  have  as-accurate  an 
idea  as  possible  concerning  tne  nature  of  the  disability,  and  its 
extent.   It  is  also  very  desirable  to  make  the  pur 11  conscious  of  the 
nature  of  his  difficulty,  and  of  what  he  must  do  to  remove  it.   lluch 
emphasis  has  been  placed  upon  this  last  point  by  tne  school  authori- 
ties at  \7innetka,  Illinois  and  Los  Angeles,  California.   In  these 
cities  it  has  been  found  very  much  worth  while  to  make  the  pupil 
thoroughly  conscious  of  the  "goal"  which  he  must  reach,  and  of  the 
process  by  which  he  can  reach  it. 

These  three  questions  should  be  U6ed  to  test  our  efforts  at 
remedial  work:- 

(1)  What  is  the  nature  of  the  difficulty? 

(2)  How  serious  i$  it? 

(?)  How  well  do  the  pupils  understand  their  own  needs? 

With  these  test  questions  in  mind,,  it  We  worth  while  to  discuss 
the  situation  in  arithmetical  reasoning  in  some  detail.   The  fre- 
quencies given  on  page  4  show  that  about  60%   of  the  wrong  answers  are 
due  to  a  failure  right  in  the  beginning  to  understand  what  is  to  be 
done.   About  20$  of  the  children  understand  what  to  do,  but  fail  to 
get  the  correct  answer  because  of  an  error  in  fundamentals .   Therefore, 
if  these  two  sources  of  trouble  could  be  entirely  removed  our  list  or 
errors  would  be  only  l/5  as  large  as  it  now  is.   This  would  mean  a 
tremendous  saving  of  time  and  energy  for  the  teacher,  and  an  untold 
amount  of  gain  on  the  part  of  the  pupils.  It  should  always  be  re- 
membered in  this  connection  that  the  discouragement  which  results  from 
:  :  j '  '•     fs.T;  I-*  wV'-i!.: nt  L'riO?.vr»«>  v. by.  tcr.d'n  to    icpr-ir  in  a.  serious 
..  J.-- j  th|  ctij.l&Sa  self -"confidence  in  general,  arid  aav  even  result  in 
a  feerjaanpni  feeling  of  incapacity  I. inferiority  complex).  Tne  rer.cval 
of  these  errors,  therefore ,  becomes  a  matter  of  major  importance  in 
-r.  . >  future  development^  of  our  methods  oi  teaching, 

^3r.;dJ.al  Suggestions  For  Those  V/ho  Totally  ~Tail  to  Comprehend 

flag  Problem 

little  is  known  concerning  a  child's  mental  reactions  in  the 
fa.ce  of  a-situation  which  he  cannot  understand.   The  safest  assumption 
at  the  present  stage  of  development  in  our  educational  psychology  is 
that  a  child  acts  quite  similarly  to  an  animal  under  similar  situations, 
A  hungry  rat  in  a  cage  trying  to  get  food  which  is  some  distance  out- 
side of  the  cage  responds  by  making  all  sorts  of  reactions  of  which  a 
rat  is  capable.   The  rat's  problem  is  to  get  food.   The  child  feels  a 
need  to  vrite  something  on  his  tablet  --  do  something  that  will  get  a 
number  that  looks  like  the  right  answer.  3oth  rat  and  child  are  blind- 
ly trying  to  satisfy  needs.   Neither  has  much  of  an  idea  as  to  how 
this  may  be  done.  Each  tries  the  thing  which  is  most  easy  for  him  to 
do,  —  the  thing  which,  in  the  past,  he  has  learned  to  do  best,  and 
each  will-  continue  to  make  this  seme  reaction  to  sinilar  situations 
unless  the  reaction  proves  unsatisfactory,   In  short,  whenever  humo>.n 
bo ings  meet  a  new  situation  which  they  cannot  analyze,  they  invariable* 
fall  back  upon  the  animal  type  of  behavior  in 


(-9) 

such  circumstances.   In  educational  psycnology  tiria  method  of  meeting 
new  situations  is  called  trial  and  error  or  fumbling  and  success. 
Our  problem' therefore  is  fco  substitute  a  higher  type  of  learning  for 
a  lower  one.   This  is  the  essence  of  teaching  p'.n-ilo  to  thins  and 
reason.  We  must  meet  this  prcblea  in   some  aaanner'. 

Freresaij  t^te s _3? or_  yhinkirg^ 

Specific  directions  on  how  to  teach  children  to  think  are  not 
yet  available,  hut  a  few  points  are  clear  enough  t o  be  stated  somewhat 
definjtely.   They  are  as  follows:* 

1.  A  supply  of  free  ideas  or  concepts  must  be  provided. 

2.  The  ability  to  compare,  combine,  and  judge  the  relative  value  of 
these  ideas  for  the  particular  situations  is  necessary, 

3.  All  of  the  relevant  ideas  must  function,  but  none  that  are  'irrele- 
vant can  enter  into  the  thinking  process. 

4.  Relevant  ideas  must  be  encouraged  and  rewarded;  those  that  are 
irrelevant  must  be  discouraged  and  penalized* 

5.  Thinking  is  impossible  until  adequate  experience  has  been  provided 
in  trial  and  error  and  imitation  methods .   The  amount  which  may 

be  called  adequate  varies  according  to  the  intelligence  and  past 
experience  of  tne  child. 

The  foregoing  may  be  regarded  as  well  established  principles 
in  teaching  of  thinking.   The  present  problem-  of  the  field  worker 
in  education  is  to  develop  a  technique  which  will  assist  in  the  real 
ization  of  the  needs  as  they  are  now  known. 

Some  things  to  do  for  children  who  show  a  total  failure  to  com- 
prehend the  problem  are  as  follows:* 

To  increase  the  supply  of  free  jdeas  or  concepts. 

Find  whether  or  not  the  child  is  reading  the  problem  correctly - 
This  can  be  done  in  a  very  satisfactory  manner  by  using  the  problems 
f'S  silent  reading  material  before  any  attempt  is  made  at  solution. 

Samples  of  silent  reading  exercj  ses  v'hich  are  _prer e qui 8 it e  to  the 

S. Qlution  of  Problems  in  Arithmetic. 

To  the  pupils  Can  you  understand  your  arithmetic  problems  when 
you  read  them?   If  so,  you  can  do  these  e3terci.se  correctly.  After 
each  problem  is  a  list  of  things  which  might  be  done.   You  are  to  show 
that  you  understand  by  drawing  a  line  under  the  word  or  words  whish 

tell  the  right,  thinp;  to  do. 

(l)  Mary  wishes  to  BS&e  35  cents;  if  she  has  already  saved  24  cents, 
how  much  more  must  she  sa.^e? 

Add         Subtract  Multiply        Divide 


-3  0- 

(2)  How  many  tablets  can  I  buy  for  40  cents,  if  each  tablet  costs 
8  cents? 

Add        Subtract  Multiply    Djvi.de 

(3)  Fred's  toother  has  3  revs  of  fruit  Jars  with  9  jars  in  each  ror. 
How  many  fruit  jars  has  she? 

Add        Subtract  Multiply    Divide 

(4)  Frank  had  2  apples  and  .his  mother  gave. him  3  more.  How  many 
apples  did  he  then  have? 

Add        Subtract  Multiply    Divide 

Fill  in  the  blanks  and  underline  the  words  which  tell  the  right 
things  to  do. 

(5)  How  much  cheaper  is  it  to  rent  a  room  for  a  year  at  $12.50  a 
month  than  it  is  to  rent  another  foam  for  the  same  length  of 
time  at  &3.50  a  week? 

..  Divide  by _ 


Divide  b; 


Multiply by 

Multiply by 

Th*n  Add     Subtract      Multiply       Divide 

(6)  A  farmer  sold  3375  bushels  of  potatoes  at  &0.90  a  bushel,  and 
wishes  to  buy  j.and  at  $75  an  acre.   How  m&£y  acres  can  5te  buy?  = 

Divide  by 


Multiply by , 

Multiply  S by 

Tner.  Add       Subtract    Multiply        Divide 

(7)  If  a  man  receives  #7. 20  for  8  hrs .  work,  how  much  should  he  get 
.  for  6  1/2  hrs..vork? 

Add     __to Multiply  by    


Subtract frorn Divide by  _ 

Then   Add  Subtract        Multiply       Divide 

Exercises  like  the  above  will  insure  against  careless  reading. 
If  the  child  fails  tc  respond  correctly  after  a  short  practice  upon 
exercises  of  this  sort,  it  may  be  taken  as  a  sure  sign  that,  another 
t/ je  of  prerequisite  training  is  necessary.   Such  children  have  not 
had  sufficient  experience  with  concrete  things.   Free  ideas  are  lack*" 
ing  and  must  be  obtained,  if  at  all,  through  the  actual  handling  of 
the  objects.   They  wj 11  need   repeated  practice  of  this  sort  which 
deals  at  first  with  one  step  problems  only. 


-11- 

In  problem  I  above  it  will  be  necessary  to  provide  toy  or  real 
money.   In  problem  2  actual  apples  «ril3i  b3  needed  and  so  on.   In" 
general  th3?e  children  wall  be  lacking  in  intelligence  also.  Much 
tjme  and  patience  will  p      ,y  b9  needed.   When  success  teOV  been 
attained  with  tne  simple  problems  more  complicated  on^s  snould  be 
used. 

Some  things  to  do  for  children  who  give  only  partial  responses  to  the 
problem. 

Most  of  the  vtrouble  here  is  due  to  a  failure  to  read  the  problem 
completely.  Exercises  like  the  following  will  prove  neipful. 

To  the  pupil:  Some  boys  and  girls  fail  to  get  their  problems  be- 
cause they  do  not  see  everything.  How  sharp  are  your  eyes?  You  will 
get  caught  in  the  following  exercises  i^  you  are  not  careful. 

(1)  A  boy  had  240  marbles  and  lost  l/4  of  them.   How  many  had  he 
left? 

Are  ycu  a3±ced  to  find  how  many  marbles  the  bey  lost? 

(2)  John  sold  his  rabbit  for  20  cents  which  was  4/5  of  what  the 
rabbit  cost  him.   How  much  did  he  pay  for  the  rabbit? 

Underline  the  words  wnich  make  the  following  statment  true. 

John  sold  his  rabbit  for  (more  thai.,  less  than)  it  cost  hime 

Are  you  asked  to  find  out  how  much  money  John  lost  on  his 
rabbit? 

Underline  the  word  which  tells  the  right  things  to  do. 

Add       Subtract  Multiply        Divide 

Some  things  to  do  for  children  who  fa.-"l  to  understand  ouanti tative 
relations. 

Make  a  list  of  the  important  quantitative  relations  which  occur 
in  the  problems  which  ycu  assign.   The  following  list  is  suggestive: 

(1)  The  relation  between  cost,  expenditure,  income,  and  selling 
price. 

A  certain  house  rents  for  $960  a  year,  the  taxes  and  uokeep  cost 
$300  a  year.   How  much  can  I  afford  to  pay  for  this  house  in 
order  to  clear  6%   a  year  on  my  investment?  Suppose  I  have 
bought  the  house  at  this  price  and  am  offered  ^12,000  for  it. 
Should  I  sell  ?  Why? 

(2)  The  relation  between  rate,  time,  and  distance. 

The  distance  from  New  York  to  Chicago  is  980  miles. 

Ho\v  long  will  it  take  a  train  to  run  this  distance  <*%    the  race 

of  49  miles  per  hour? 

(3)  The  relation  between  length,  breadth,  and  thickness. 


-12- 

I  need  a  bin  that  will  contain  <M0  cubic  feet*   if  I  maka  it  8 
feet  long  and  5  feet  wide,  how  deep  must  it  te? 

(4)  Relations  involving  proportion. 

It  takes  9  men  5  days  to  build  a  piece  of  road.   How  many  .nen 
would  it  take  to  build  this  road  in  3  days? 

Problems  of  this  sort  are  of  great  practical  value  and  are  a 
serious  source  of  error  in  both  Arithmetic  and  Algebra.,  The  best 
method  of  teaching  them  is  not  known.   It  is  the  belief  of  the  writer 
that  much  help  can  be  gotten  for  this  type  or  difficulty  by  teaching 
the  equation  in  arithmetic.   Simple  equationa  are  net  beyond  the  com- 
prehension of  most  elementary  school  children   Some  children  will  net 
understand  them,  but  tnese  children  will  net  understand  ary  method. 
Even  with  them  the  equation  form  is  desirable  because  it  can  be  taught 
in  a  mechanical  form,  the  meaning  of  which  the  child  will  appreciate 
later.   The  following-  equations  relate  to  the  four  problems  given  afeo^re . 
Only  the  first  letters  of  the  words  are  u^ed. 

1.  I  (income)   -  E  (expenditure)  =  C  (amount  cleared) 
C  (cost)  -£- G(gain)  »  S  (sell'5  ng .  pr  i  ce  ) 

2.  D  (distance)  r  R  (rate)x  T  (time) 

3.  C  (contents)*  L  (length)  x  B  (breadth)  x  D  (depth) 

4.  9x5=   ?x3 

In  case  G  is  wanted  as  in  dumber  1.  start  with  C  4r  G  -  S.  Then 
teach  that  G  which  is  net  wanted  on  the  same  side 'with  G  can  be  put 
over  with  S  by  charging  its  sign  as  in  G  =  S  -  C. 

In  case  D  is  wanted  as  in  Number  3,  start  with  G  =.  L  x  B  x  D. 
Then  teach  that  D  can  be  made  to  stand  alone  by  putting  L  and  E  in  thS 
other  side  and  changing  their  signs  as  in  C  ■?  L  i  3.  ■  D. 

"Some  tningp  to  do  for  children  who  make  errors  in  the  fundamental 
processes. 

Tnis  topic  is  discussed  in  a  circular  of  this  department  entitled; 
"A  Study  Of  Errors  In  Arithmetic" ,  and  will  be  treated  further  in  a 
series  of  circulars  on  " Inventory!}',  and  "Reaidial  Exercises  In  Arith- 
metic", which  are  now  in  preparation. 


UNIV 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


